Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

int main() {
	int n;
	scanf("%d", &n);
	vector<double> f(n+1, 0);
	f[0]=1;
	for (int i=1; i<=n; i++) {
		double p;
		scanf("%lf", &p);
		vector<double> nf(n+1, 0);
		for (int j=0; j<=i; j++) nf[j]=(j>=1?f[j-1]*p:0)+f[j]*(1-p); 
		f=nf;
	}	
	double ans=0;
	for (int i=n/2+1; i<=n; i++) ans+=f[i];
	printf("%.10f\n", ans);
	return 0;
}